The increasing integration of renewable energy sources results in fluctuations in carbon intensity throughout the day. To mitigate their carbon footprint, datacenters can implement demand response (DR) by adjusting their load based on grid signals. However, this presents challenges for private datacenters with diverse workloads and services. One of the key challenges is efficiently and fairly allocating power curtailment across different workloads. In response to these challenges, we propose the Carbon Responder framework. The Carbon Responder framework aims to reduce the carbon footprint of heterogeneous workloads in datacenters by modulating their power usage. Unlike previous studies, Carbon Responder considers both online and batch workloads with different service level objectives and develops accurate performance models to achieve performance-aware power allocation. The framework supports three alternative policies: Efficient DR, Fair and Centralized DR, and Fair and Decentralized DR. We evaluate Carbon Responder polices using production workload traces from a private hyperscale datacenter. Our experimental results demonstrate that the efficient Carbon Responder policy reduces the carbon footprint by around 2x as much compared to baseline approaches adapted from existing methods. The fair Carbon Responder policies distribute the performance penalties and carbon reduction responsibility fairly among workloads.